When providing healthcare to patients it is frequently important to accurately monitor at least one type of parameter associated with the patient. To accomplish this, at least one sensor is connected to a patient for use in sensing physiological signals that are provided to and interpreted by at least one type of patient monitoring device. The sensed physiological signals are used in determining the at least one patient parameter. Sensed signals having poor quality (e.g. interference from external sources) negatively impact the ability of the patient monitoring device to determine the desired patient parameter resulting in inaccurate patient parameter data values. Another problem associated with data obtained from a signal having poor quality relates to the use of inaccurate data for diagnostic purposes. Inaccurate patient parameter data derived from a signal having poor quality increases the likelihood of a false positive indication of a particular medical condition. To remedy these drawbacks, an adaptive notch filter has been developed and implemented in patient monitoring devices. A notch filter can effectively remove power line interference through a learning process whereby a step size of the notch filter is set enabling the notch filter to filter out the undesired portion of the input signal (e.g. external powerline interference). The step size is crucial to the performance of an adaptive algorithm. For example, a small step size rarely causes divergence, but takes long time to converge and a large step size converges quickly, but may cause divergence. Alternatively, using the larger step may also result in what is known as a “ringing artifact”. It is therefore desirable to minimize ringing artifact while maximizing convergence of a filtered input signal.
FIG. 1 is a prior art block diagram of a conventional notch filter 100 that may be used to remove powerline noise from an input signal. The notch filter 100 may include a processor 102 that executes an adaptive algorithm that selectively estimates an amount of interference on an input signal x(n). The algorithm controls a summing function 104 to automatically filter the input signal x(n) by a certain value thereby removing the estimated interference therefrom. In operation, the primary input signal x(n) is the combination of the interested signal and interference. The processor 102 executes an adaptive algorithm to determine data representing an amount of estimated interference w(n) that is present in input signal x(n). An exemplary adaptive algorithm executed by the processor 102 may be found below in Table 1.
TABLE 1Notch Filter Pseudo-codeW1w(n − 1) from previous iterationw2w(n − 2) from previous iterationy1y(n − 1) from previous iterationfc = cos(2*pi*f0/fs)f0 is the notch frequencyfs is the sampling ratefor n = 1:NLoop through all data samplesy0 = x(n) − w1Input minus oscillation signaly(n) = y0Save filtered results to the output bufferd = y0 − y1Calculate errorz = f(d)Find step size. It is a function of d.w = 2*fc*w1 − w2 + zPredict oscillator signalw2 = w1Shift oscillator samplesw1 = wShift oscillator samplesy1 = y0Shift output samplesendEnd of the for-loop
Upon determining the estimated interference w(n), the processor 102 provides the value w(n) to a negative input of the summing function 104. The summing function 104 automatically filters the subsequent input signal x(n) to remove the estimated interference from the input signal x(n) in order to generate an output signal y(n) which optimally only includes the interested signal. Thus, output signal y(n) represents the difference between input signal x(n) that includes both the interested signal and the interference and the estimated interference w(n) as determined by the processor 102. The processor 102 is able to selectively and continually adjust the value of estimated interference w(n) to reflect the optimal estimation of the interference to be removed from the input signal x(n).
In previous iterations, the notch filter may be a phase-locked-loop (PLL) filter. However, the adaptive filter may operate on an input signal without the use of a reference signal typically required by a conventional adaptive filter. The cost function of the adaptive filter may be represented by the equation as shown in Equation 1.min[y(n)−y(n−1)]2  (1)Furthermore, the adaptive algorithm implemented adjusts the oscillation signal w(n) at each step as shown in Equation 2 which states
                                          w            ⁡                          (                              n                +                1                            )                                =                                    2              ⁢                                                          ⁢                              cos                ⁡                                  (                                      2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                  f                        0                                                                    f                        s                                                                              )                                            ⁢                              w                ⁡                                  (                                      n                    -                    1                                    )                                                      -                          w              ⁡                              (                                  n                  -                  2                                )                                      +                          μ              ⁡                              (                                                      y                    ⁡                                          (                      n                      )                                                        -                                      y                    ⁡                                          (                                              n                        -                        1                                            )                                                                      )                                                    ,                            (        2        )            where μ controls the step size and μ>0, f0 is the notch frequency, fs is the sampling rate, and n is the sample index. Equations 1 and 2 explain how the adaptive algorithm may be implemented in a self-referencing adaptive notch filter.
For example, according to Eq. (1), the cost function is min[y(n)−y(n−1)]2 and the gradient of the cost function reflects the steepest ascent of the cost function. Thus, as y(n)=x(n)−w(n), the gradient of the cost function with respect to w(n) can be written in accordance with Equation 3 which states:
                                                                                                              d                    ⁡                                          [                                                                        y                          ⁡                                                      (                            n                            )                                                                          -                                                  y                          ⁡                                                      (                                                          n                              -                              1                                                        )                                                                                              ]                                                        2                                                  d                  ⁢                                                                          ⁢                  w                                            =                            ⁢                                                                    d                    ⁡                                          [                                                                        x                          ⁡                                                      (                            n                            )                                                                          -                                                  w                          ⁡                                                      (                            n                            )                                                                          -                                                  y                          ⁡                                                      (                                                          n                              -                              1                                                        )                                                                                              ]                                                        2                                                  d                  ⁢                                                                          ⁢                  w                                                                                                        =                            ⁢                              -                                  2                  ⁡                                      [                                                                  y                        ⁡                                                  (                          n                          )                                                                    -                                              y                        ⁡                                                  (                                                      n                            -                            1                                                    )                                                                                      ]                                                                                                          (        3        )            In order to find the minimum of the cost function we need to take a step in the opposite direction of the gradient which is expressed mathematically as Equation 4 which states:
                                                                        w                ⁡                                  (                                      n                    +                    1                                    )                                            =                            ⁢                                                w                  ⁡                                      (                    n                    )                                                  +                                  a                  ⁡                                      (                                          -                                                                                                    d                            ⁡                                                          [                                                                                                y                                  ⁡                                                                      (                                    n                                    )                                                                                                  -                                                                  y                                  ⁡                                                                      (                                                                          n                                      -                                      1                                                                        )                                                                                                                              ]                                                                                2                                                                          d                          ⁢                                                                                                          ⁢                          w                                                                                      )                                                                                                                          =                            ⁢                                                w                  ⁡                                      (                    n                    )                                                  +                                  2                  ·                  a                  ·                                      [                                                                  y                        ⁡                                                  (                          n                          )                                                                    -                                              y                        ⁡                                                  (                                                      n                            -                            1                                                    )                                                                                      ]                                                                                                          (        4        )            where a is the step size and a>0. Additionally, μ=2a and thus can be rewritten as Equation 5 which states:w(n+1)=w(n)+μ·[y(n)−y(n−1)]  (5)Moreover, when w(n)(estimated interference) is a pure sinusoid signal, it can be written as Equation 6 which states
                                          w            ⁡                          (              n              )                                =                      A            ⁢                                                  ⁢                          sin              ⁡                              (                                                      2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                  f                        0                                                                    f                        s                                                              ⁢                    n                                    +                  φ                                )                                                    ,                            (        6        )            where A is the amplitude, f0 is the frequency, fs is the sampling rate, and φ is the phase. Thus, according to Eq. (4), w(n−1) and w(n−2) can be written as Equations 7 and 8, respectively, which state
                              w          ⁡                      (                          n              -              1                        )                          =                  A          ⁢                                          ⁢                      sin            ⁡                          (                                                2                  ⁢                                                                          ⁢                  π                  ⁢                                                            f                      0                                                              f                      s                                                        ⁢                                      (                                          n                      -                      1                                        )                                                  +                φ                            )                                                          (        7        )                                          w          ⁡                      (                          n              -              2                        )                          =                  A          ⁢                                          ⁢                      sin            ⁡                          (                                                2                  ⁢                                                                          ⁢                  π                  ⁢                                                            f                      0                                                              f                      s                                                        ⁢                                      (                                          n                      -                      2                                        )                                                  +                φ                            )                                                          (        8        )            
If
      α    =                  2        ⁢                                  ⁢        π        ⁢                              f            0                                f            s                          ⁢                  (                      n            -            1                    )                    +      φ        and            β      =              2        ⁢                                  ⁢        π        ⁢                              f            0                                f            s                                ,  then w(n), w(n−1), and w(n−2) can be written as Equations 9-11, respectively, as follows:
                              w          ⁡                      (            n            )                          =                              A            ⁢                                                  ⁢                          sin              ⁡                              (                                                      2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                  f                        0                                                                    f                        s                                                              ⁢                    n                                    +                  φ                                )                                              =                      A            ⁢                                                  ⁢                          sin              ⁡                              (                                  α                  +                  β                                )                                                                        (        9        )                                          w          ⁡                      (                          n              -              1                        )                          =                              A            ⁢                                                  ⁢                          sin              ⁡                              (                                                      2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                  f                        0                                                                    f                        s                                                              ⁢                                          (                                              n                        -                        1                                            )                                                        +                  φ                                )                                              =                      A            ⁢                                                  ⁢                          sin              ⁡                              (                α                )                                                                        (        10        )                                          w          ⁡                      (                          n              -              2                        )                          =                              A            ⁢                                                  ⁢                          sin              ⁡                              (                                                      2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                  f                        0                                                                    f                        s                                                              ⁢                                          (                                              n                        -                        2                                            )                                                        +                  φ                                )                                              =                      A            ⁢                                                  ⁢                          sin              ⁡                              (                                  α                  -                  β                                )                                                                        (        11        )            By using the following trigonometric identity of Equation 12,
                                          sin            ⁡                          (              x              )                                +                      sin            ⁡                          (              y              )                                      =                  2          ⁢                                          ⁢                      sin            ⁡                          (                                                x                  +                  y                                2                            )                                ⁢                      cos            ⁡                          (                                                x                  -                  y                                2                            )                                                          (        12        )            the result is shown in Equation 13 which statesw(n)+w(n−2)=2A sin(α)cos(β)=2 cos(β)w(n−1).  (13)And Equation 13 may be rewritten as follows in Equation 14
                                                                        w                ⁡                                  (                  n                  )                                            =                            ⁢                                                2                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                      β                      )                                                        ⁢                                      w                    ⁡                                          (                                              n                        -                        1                                            )                                                                      -                                  w                  ⁡                                      (                                          n                      -                      2                                        )                                                                                                                          =                            ⁢                                                2                  ⁢                                                                          ⁢                                      cos                    ⁡                                          (                                              2                        ⁢                                                                                                  ⁢                        π                        ⁢                                                                              f                            0                                                                                f                            s                                                                                              )                                                        ⁢                                      w                    ⁡                                          (                                              n                        -                        1                                            )                                                                      -                                  w                  ⁡                                      (                                          n                      -                      2                                        )                                                                                                          (        14        )            By replacing the value of w(n) in Eq. (5) with Eq. (14), the result is shown in Equation 15 which states that
                              w          ⁡                      (                          n              +              1                        )                          =                              2            ⁢                                                  ⁢                          cos              ⁡                              (                                  2                  ⁢                                                                          ⁢                  π                  ⁢                                                            f                      0                                                              f                      s                                                                      )                                      ⁢                          w              ⁡                              (                                  n                  -                  1                                )                                              -                      w            ⁡                          (                              n                -                2                            )                                +                                    μ              ⁡                              (                                                      y                    ⁡                                          (                      n                      )                                                        -                                      y                    ⁡                                          (                                              n                        -                        1                                            )                                                                      )                                      .                                              (        15        )            Thus, as Equation (15) is equivalent to Equation (2), an adaptive algorithm may be implemented in a notch filter that does not include a reference signal, such as the one shown in FIG. 1.
While adaptive notch filters have had some success in estimating an amount of interference in an input signal to provide a filtered signal, these filtered signals often time have undesirable characteristics associated therewith resulting from less than optimal step size used by the notch filter. A system according to invention principles addresses deficiencies of known systems.